 No entire square root of the identity function

If an entire function $h$ satisfied $h(z{)}^2=z$, then $h(0)=0$. Differentiating would give $2h(0)h^{\prime }(0)=1$, a contradiction. More generally, the order of every zero of a square of a holomorphic function is even, so a simple zero prevents a holomorphic square root on any neighborhood containing it. A square root can still exist on a slit domain avoiding the zero. The identity theorem can force this obstruction from values specified only on a sequence accumulating elsewhere in a connected domain.